Aircraft: B737
Phase of flight: Climb
Anomaly: Aircraft Equipment Problem Less Severe; Deviation - Track / Heading All Types

Aircraft navigation was severely degraded due to multiple MELs. LNAV/VNAV; autothrottles; FO flight director and numerous other items were MELed. Before flight; Captain had called Dispatch to ask if we could swap aircraft. We were told there were no spares in ZZZ or ZZZ1 and to continue operation of that aircraft.Due to limitations; Dispatch supposedly filed our flight /L for degraded NAV capability. Enroute from ZZZ2 to ZZZ3; Center gave us direct ZZZZZ intersections. We put it in the FMC and began navigating towards ZZZZZ intersection via heading. Center called us and asked if we showed on course and we did. No other conversation or communication was said about our flight path. This ASAP should not have to be written. I place most of the blame on myself; but [company] [and Maintenance Controller] should never have allowed this aircraft on the line; in my opinion unless; it was going to fly in non-congested airspace between non-congested airports with minimal enroute flying times of 1.5 hours or less. We flew the jet from ZZZ-ZZZ1-ZZZ2-ZZZ3. I should never have agreed to fly this crippled aircraft for a third leg. I asked Dispatch repeatedly for an aircraft swap; beginning [on leg 1]; and Dispatch repeatedly told me that nothing was available for a swap; even though we went through two maintenance bases. Take a look at our Release; then it will become abundantly clear why I say this. In 13 years; I have never seen a list of MELs like this on any [of our older aircraft]; let alone a 737-NG. It took me 30 minutes to simply read through and verify all of the MELs. My FO and I were in the 'red' and we hadn't even left the gate [for leg 1]. After talking with the outgoing Capt; however; reading and verifying all of the MELs; and getting some assurance from Dispatch that they would look for ways to swap the aircraft out downline; we were able to bring ourselves out of the 'red'. After agreeing that I would handle the PF duties and thoroughly briefing and understanding the aircraft's limitations; we pushed on our first leg to ZZZ1. The short leg to ZZZ1 was uneventful. Even though LNAV was MEL'd because of the MCP; we still had the ability to back-up our VOR to VOR navigation with the magenta line on the ND. It was mentally fatiguing switching back and forth between VOR and MAP on the EFIS and ND; and between HDG SEL and VOR/LOC on the MCP; but it was doable. In fact; the confidence we gained on that relatively short and benign leg contributed; I'm sure; to the mistakes we would make 2 legs and several hours later; when we would be in a much more dynamic and challenging environment going from ZZZ2-ZZZ3 in Center's airspace. Enroute to ZZZ2; I sent another ACARS message to Dispatch inquiring about a possible aircraft swap upon arrival. In so many words; the dispatcher said that they had checked; but nothing was available. Upon arrival in ZZZ2; I think it's safe to say that; at this point; my patience was wearing thin--having just nursed this jet for another 2.5 hours and then was told; once again (at another maintenance base); that an aircraft swap was not available. In the 3 years I have been a Captain; I have never rejected an aircraft. However; if I had to do this all over again...given the mental fatigue I had experienced to that point; I can honestly say I will never do that again! I should have called the [Chief Pilot] right then; but didn't (mistake 1). Although frustrated and tired; we only had one more leg to go. We had also become quite proficient at flying this navigation-limited aircraft so; with the concurrence of my FO; we decided we could do one more leg.I felt the background above; though lengthy; was necessary to understanding what happened next but; more importantly; to paint a picture and hopefully provide feedback to those at the [Maintenance Controller] who are making the decisions to let an aircraft like this one fly all day 'on the line.' Its one thing to fly ZZZ-ZZZ1 or [other small cities we serve]; etc. It's quite another when you are tired and frustrated to fly in that busy and congested airspace.My frustration with not getting another aircraft was compounded even more when I received the Release and it reflected an RNAV departure and arrival. After two calls to the dispatcher and three releases later; we finally got a release and a flight plan that we could legally fly. It was a simple flight plan. I was familiar with [two of the three VORs]. I looked on the High chart; but couldn't find the third VOR and assumed it was a low VOR. Since it was in our data base; I felt comfortable and continued with flight preparation and our briefings. Upon initial contact with Departure; they immediately cleared us direct to ZZZZZ. My FO told them we were unable and needed direct ZZZ2 [VOR]. The controller then asked why we couldn't go to ZZZZZ intersection since it was on the SID. This is where my impatience clicked in. Since we had a magenta line and good FMCs and NDs; I just told my FO to accept direct. I was not in a mood to argue and just wanted to get out of ZZZ2 airspace at that point. It didn't get any better after that though; unfortunately. I don't know if the Release was annotated incorrectly with respect to our navigation capability; or if both Departure and Center's expectation bias was such that they simply automatically assumed we were LNAV capable. In any case; they both kept clearing us direct to waypoints when we were not technically allowed to do that. B737NG flight crew reported being dispatched in an aircraft with numerous MELs including no LNAV; autothrottles; and the First Officer's flight director. Company was unable to swap them to a different aircraft; so they flew 3 legs and became fatigued due to the additional work of navigating VOR to VOR in challenging airspace.